A086792 Orders of finite groups G with the property that the sum of the orders of all the proper normal subgroups of G equals the order of G.

6, 12, 28, 30, 56, 360, 364, 380, 496, 760, 792, 900, 992, 1224, 1656, 1680, 1980

Offset

1,1

Comments

The only Abelian groups with this property are the cyclic groups C_n where n is a perfect number, so this sequence can be seen as a groups analogy of perfect numbers.

Derek Holt (mareg(AT)mimosa.csv.warwick.ac.uk) computed the orders of the non-Abelian groups in the sequence up to n=500 and commented "In general, if 2^n - 1 is a Mersenne prime, then 2^(n-1)*(2^n - 1) is a perfect number and the group with presentation < x,y | x^(2^n-1) = 1, y^(2^n) = 1, y^-1 x y = x^-1 > has order equal to the sum of the orders of its proper normal subgroups." So if n is an even perfect number, 2n also belongs to this sequence (the numbers 12 and 56 above).

Links

Table of n, a(n) for n=1..17.

Tom Leinster, Perfect numbers and groups, arXiv:math/0104012 [math.GR], 2001.

Tom De Medts and Attila Maróti, Perfect numbers and finite groups

Tom De Medts, Leinster groups. [archived]

Sci.math, Groups analogy of perfect numbers

Crossrefs

Subsequence of A060652.

Cf. A000396.

Sequence in context: A338520 A339472 A348034 * A064987 A057341 A068412

Adjacent sequences: A086789 A086790 A086791 * A086793 A086794 A086795

Keyword

nonn,more

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 04 2003

Extensions

a(10)-a(17) added using "Leinster groups" link by Eric M. Schmidt, May 02 2014

Status

approved